{"paper":{"title":"Average number of zeros and mixed symplectic volume of Finsler sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Boris Kazarnovskii, Dmitri Akhiezer","submitted_at":"2018-02-08T08:14:43Z","abstract_excerpt":"Let $X$ be an $n$-dimensional manifold and $V_1, \\ldots, V_n \\subset C^\\infty(X, \\mathbb R)$ finite-dimensional vector spaces with Euclidean metric. We assign to each $V_i$ a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle of $X$. We prove that the average number of isolated common zeros of $f_1 \\in V_1, \\ldots, f_n \\in V_n$ is equal to the mixed symplectic volume of these Finsler ellipsoids. If $X$ is a homogeneous space of a compact Lie group and all vector spaces $V_i$ and their Euclidean metrics are invariant, then the average numbers of zeros satisfy "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.02741","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}